My not-so-profound thoughts about valuation, corporate finance and the news of the day!

Monday, February 28, 2011

Buffett and Black-Scholes

As always, I am playing with fire when I critique Warren Buffett, but he does indulge in hyperbole (I hope that is all it is..) when he strays from his preferred habitat. In fact, my previous post on him evoked some strong responses. In the last Berkshire Hathaway report, he is quoted as saying “Both Charlie and I believe that Black-Scholes produces wildly inappropriate values when applied to long-dated options… Academics’ current practice of teaching Black-Scholes as revealed truth needs re-examination. For that matter, so does the academic’s inclination to dwell on the valuation of options. You can be highly successful as an investor without having the slightest ability to value an option”.

Let's take apart this statement:
a. "Both Charlie and I": I presume that Charlie here stands for Charlie Munger, the other fount of wisdom from Omaha. I guess this is supposed to add to the intimidation factor. If Charlie Munger agrees with Warren Buffett, what right-minded person would disagree, right? Charlie Munger has a way with words (I especially love this quote: "If the only tool you have is a hammer, everything starts to look like a nail.") But so do Yogi Berra and Lady Gaga, and I am not listening to investment advice from either one..

b. "Black-Scholes produces wildly inappropriate values when appled to long-dated options": So, the Black-Scholes that Mr. Buffett must be referencing must be the original Black-Scholes, with no dividend or dilution adjustments to value European options? And what exactly are these long dated options that are being valued? Warrants or management options? Since US companies are light users of the former, I would assume that it is the latter, which are not traded. If they are not traded, two questions:

i. How would Mr. Buffett know that they are wildly inappropriate? Because the values he got from these options were higher than Mr. Buffett's gut said that they should be worth? Perhaps, he should take a look at LEAPs (long term call and put options) traded on US stocks on the exchanges. As the value guru, he may think that all of these options are being over valued. If so, I would welcome his intervention in these markets.

ii. What exactly did Mr. Buffett do with the Black-Scholes model? The Black-Scholes model is only as good as its inputs. With long term options, the variance that should be used in the model is a long term variance (which may be well below the current level) and if the options are management options, you should be correcting for dilution and illiquidity. Since FASB has required companies to value management options and expense them for the last three years, this is a well researched area of finance. With the adjustments, the Black Scholes delivers reasonable values for options.

Here is the bottom line. The Black Scholes under values deep out of the money options (because of its assumption that prices move continuously) and over values options that are illiquid. To compensate, we can either modify the Black Scholes or use a binomial option pricing model, both of which deliver much better estimates of option value than any individual's gut...

c. "Academic’s inclination to dwell on the valuation of options": I love this one! Where does Warren Buffett's academic live? Is he a Phd student that Buffett and Munger trapped in the 1970s and put in a hut in Omaha, poring over old Journals of Finance (preferably from the 1960s)? That may explain Buffett's fixation with the CAPM and the Black Scholes. There are some academics and many practitioners who dwell on the valuation of options, but there is a reason for that. It is their job is to assess the value of listed options, warrants or convertibles, and if that is their job, they have to just dwell on the valuation of options. I am an academic (in Buffett's sense of the word) but option valuation is an after thought to me, not a central part of either corporate finance and valuation..

d. "Academics’ current practice of teaching Black-Scholes as revealed truth needs re-examination: I would be interested in what constitutes "current" practice to Mr. Munger and Mr. Buffett. Furthermore, what are these nasty academics doing? Are they telling their students to value options using the Black-Scholes model, to buy under valued options and sell over valued options?

e. "You can be highly successful as an investor without having the slightest ability to value an option": Here is the only statement that I completely agree with. Absolutely, but only if you stay away from option laden investments (which includes companies like Cisco which have a significant management option overhang, oil companies with undeveloped reserves like Petrobras, pharmaceutical companies with potential blockbuster drugs making their way through the pipeline).

I am sorry if you find me to be disrespectful for not treating Warren Buffett as a minor deity, whose every word is gospel. It is clear that all of the praise that he receives from his followers has gone to his head. He sound absurd when he talks about derivatives and seems to think that he is a macro forecaster (which actually cuts against everything he stood for two decades ago). I will pay him the ultimate compliment (or insult) by taking every macro suggestion that he makes and doing the opposite.

Options are just tools to evaluate underlying assets contingent to a specific event, or viceversa to understand the probabilities implied in the price that an event will happen.

The strength of the results is directly correlated to the quality of the inputs used in the model.

I'm convinced that part of the investment community really doesn't understand the logic behind the model or the inputs used, creating gross misjudgments over valuations (actually a good opportunity for risk arbitragers). But this doesn't mean that the tool is flawed.

I think the long-dated options Buffett is referring to are the ones he first mentioned in 2006 and mentioned in more detail in 2007:

“The puts in these contracts are exercisable only at their expiration dates, which occur between 2019 and 2027, and Berkshire will then need to make a payment only if the index in question is quoted at a level below that existing on the day that the put was written.”

and again in 2008:

"Our put contracts total $37.1 billion (at current exchange rates) and are spread among four major indices: the S&P 500 in the U.S., the FTSE 100 in the U.K., the Euro Stoxx 50 in Europe, and the Nikkei 225 in Japan. Our first contract comes due on September 9, 2019 and our last on January 24,2028. We have received premiums of $4.9 billion, money we have invested."

These are very long-dated options with 12-20 years until expiration when they were purchased. A look at the historical growth of these indexes over similar time periods in the past would enable Buffett to calculate a range of possible scenarios for the future. There are some, but not a lot, of 12 year to 20 year periods with negative returns (the Nikkei from it's highs in the late 80's and early 90's being the one such case). If Buffett is able to successfully avoid investing in these put contracts at times of abnormally high market valuations, the probability of a loss over the life of the options is very low. Buffett received $4.9 billion on a notional value of about $37.1 billion so the indexes can decline an average of 13% and he will break even.

However, there is an additional source of value in these long-dated put options -- Buffett is able to invest the premiums. If he can compound the premiums at an average of 8% per year for the 16 year average life of the contracts his return on the premiums alone will be $12.1 billion. The return plus the premium earned means the indexes can drop about 45.3% and he will break even.

I agree with Buffett on this one, and if I had access to 12+ year options I would do the same thing in a heartbeat. I haven't found any mispricings like this with LEAPs however, and I don't think Buffett means to include these in his use of the term "long-dated options".

Also, I don't think this is a macro play on Buffett's part. The indexes are made up of companies with real earnings power and these earnings will add value over the 12-20 year life of the options -- and should be reflected in the price of the indexes.

Harrison,That is a fair point. In the aftermath of crisis (any one, not just 2008) people over estimate expected volatility. That translates into high premiums for put ootions. Buffett and others made a volatility play when they sold these put options and that play has paid off handsomely. If Buffett's point was about investors over reacting to crises and misestimating volatility, I would have had no issues with it. But he managed to convert that into a swipe at the Black Scholes and academics, when neither had much to do with that mispricing.

In a 2009 article, Bradford Cornell from California Institute of Technology tries to analyse Buffett's allegations regarding the valuation of long-dated options ("Warren Buffett, Black-Scholes and the Valuation of Long-dated Options", downloadable on the internet).Unfortunately, the bottom line is not quite clear since Buffett's letter isn't precise enough and Cornell's reasoning is therefore based on assumptions as to what Buffett really meant to say. I personally think Buffett has a problem with the formula itself and not its inputs, for whatever reason.

Please actually READ Buffett's full letter before you write something like this. Your ignorance of the facts, and the points Buffett is trying to make, are just painful to read. You look exactly like the clueless academic he so often mocks.

Admin,Thank you for your incisive comment. Well.. I guess I should be gratified. I would rather be a clueless academic than a mindless zombie.

Seriously, though, I did read through Buffett's letter and I am still not getting the clue. Was it the part about his "itchy trigger finger"? (Really.. An 80-year old with his finger on the trigger of a gun gives me only one clue.. Turn tail and run).

Was it the part about his belief that this was the time for infrastructure investment? (Sounded like a front man for the Obama administration on that one).... Perhaps, this has ties to the part where he boasts about how much money he has made on his Goldman investment because the powers that be prevented Goldman from settling accounts?

Was it the fact that Berkshire Hathaway has lagged the S&P 500 since 2008? But that cannot be a clue or even true.. Buffett always beats the market)..

So, I guess I will stay clueless while those enlightened ones who got the hidden message from the letter make millions this year.

Err, sorry. That should have read:"as a result any put option for very long time horizons priced using Black-Scholes will be too expensive."

Buffett probably also has issues with the implicit assumption of stock returns averaging at the risk-free rate over long periods that stems from the no-arbitrage condition (difficult to justify given the frictional costs of dynamic hedging for 10 years) and consequent risk-neutral pricing for options, but that's just my guess.

All that said, he definitely takes swipes at a straw-man academic each year. It's part of the folksy persona that helps him net the sweet deals, so pretty easy to understand.

Warren Buffett has been critical of academic theories for a long time. and its natural for the academia to hate him for it. This post looks more like a desperate attempt to hit back at him than an honest critique. I mean, you even find fault with "Both Charlie and I"??? That's really lame. I don't expect you to worship Buffett. you are welcome to criticize him, but i would expect a respected figure like you to stick to intelligent reasoning rather than rhetoric.

and by the way, people don't listen to charlie munger just because of his witticisms. while yogi berra's witticisms only serve to entertain, charlie munger's convey practical wisdom. warren buffett recognized this and has benefited hugely. it makes sense to listen to him and learn rather than dismissing him off as just someone who "has a way with words"

Silent Observer,My problem with Buffett and Munger (and this has become particularly true in the last few years) is that their investment justification has become "trust us. we know what we are doing and we don't have to tell you why". So, when Buffett's primary justification for selling long term put options is that he and Munger think that the model yields values too high, I will demur. They are both smart people with distinguished track records but that is thin gruel to base an investment strategy on.In fact, if you read through the entire letter, this theme seems to persevere. Invest in infrastructure and you are going to earn high returns (why now and where will the high returns come from), railroads are good investments (is it because they own valuable real estate or because they see the demand for rail transportation increasing), invest in the US (because human beings are endlessly creative and will find a way out of this mess).I thought that a central theme of value investing is that you keep your eyes on the ball, talk about specifics within companies and don't get carried away with "big" stories.

I think Buffett's point was that the B-S model gives users a sense of security (in many people's minds, complexity = validity), and that because of this false sense of security, people make irrational decisions based on faulty assumptions. So I actually don't see why Prof. Damodaran would take issue with Buffett's comments, given that both believe complex models to be, at its core, "garbage in, garbage out."

Also, Buffett's never been all that forthcoming with his reasons for specific investments. Early on in his career (according to both his major biographies, The Snowball and the Lowenstein one from the 90s), Buffett refused to allow his limited partners to inquire about specific investments. So that really hasn't changed lately.

Finally, while it's true that Berkshire Hathaway has lagged the market since 08, it performed admirably back in 07. Also, as the size of his fund has grown, it's harder for Buffett to find opportunities that others miss (as would be in the case of smaller cap companies).

and your argument that "It is their job is to assess the value of listed options, warrants or convertibles, and if that is their job, they have to just dwell on the valuation of options." only exposes the uselessness of such academicians. they are not doing it because it is of any practical use... they are doing it merely because "it is their job"buffett is absolutely right in criticizing them

I don't know how Prof Damodaran has time for making fun of Buffet and arguing with the people posting comments.

The original Black-Scholes model assumed volatility is constant independent of strike and expiration and this is known to be incorrect.

Nowadays, people relax the model to allow volatility to be dependent on time, on strike and on expiration, and this makes the "model" always correct by definition, because you can always come up with a number for the volatility that "predicts" a given option price at a given time.

So now it is not anymore a model wtih predictive power, but rather a "languange" where options can be quoted in terms of the associated volatility instead of their price.

The real question raised by Buffet is if the volatility for options expiring in 10 years should be valuated higher or lower or approx the same as short-term expiry ones.

That's a good question.Over a decade, the non-martingale statistical properties of the markets will be enhanced.

On one hand, the probability of Black swans supports higher volatilities in the long term.This means that there is "fat-tail" probability that markets will jump in price several standards of deviations either up or down due to rare unforeseeable events.

On the other hand, the "return-to-the-mean" forces will reduce the volatility.I think Buffet tends to believe that even thought the market may have a lucky/unlucky streak for several years, it cannot sustain this situation for long no matter how lucky because external constraints based on human society sets-in.

For example, oil companies can get lucky a few years if oil continues to go up, but it cannot go to infinity because alternative sources will eventually be adopted and the demand will eventually drop faster than the expected demand curve existing now. Or for example one single company, no matter how lucky they get, cannot grow its value to infinity because new regulations will be created to limit its growth, etc.

So I tend to agree with Buffet, although it is a good question for research. Unfortunately, it seems I am the only commenter here discussing it.

Javier,Now we are talking substance. I agree with you that the Black Scholes, with its assumptions, can seriously misvalue long term options and that you may want to abandon it or even shift to a different model. My issues with Buffett on the topic are:1. Buffett makes no substantive argument for why Black Scholes is wrong or what he would replace it with. Instead, he asks us to trust him and Charlie Munger in their view on this topic and their way of valuing options (which is unspecified)2. Even if we accept the premise that Black Scholes misvalues options, Buffett goes further and argues that it under values long term options. I don't see why.3.Why the ad hominem attack on academics? If Buffett believes that the buyers of options are paying too high a price, that is fine but none of these buyers are academics. So, why bring them into the picture?

So, if we want to have a debate about how best to value long term options, let's have it. But Buffett does not provide that argument.

I have found that Black-Scholes' normal distribution assumption is the root of the problem when valuing long-dated options. By using a customized probability-weighted binomial option pricing tree, you will get more accurate pricings on long-dated options. I have only recently discovered this a couple of years ago and have used this approach to consecutively make over 100% each year trading mispriced options.

The stock market over a 12-20 year period (which Buffets puts are written for) does not exhibit a normal distribution because it has an upward bias. Therefore, Buffet is potentially correct in saying that a Black-Scholes model incorporating dividends could still be priced incorrectly.

Cornwall Capital made phenomenal returns from trading long-dated options as well. They were written about in "The Big Short."

I think Black-Scholes provides an excellent approximation for most option contracts, however, it is biased for long-dated contracts unless the underlying securities revert back to a mean reverting level without an upward or downward bias.

After reading this article, I was motivated to implement the Black-Scholes model in a spreadsheet so I could play about with the parameters - it's here: http://optimizeyourportfolio.blogspot.com/2011/05/black-scholes-option-pricing-and-greeks.html

You really have misrepresented the point made by Buffett about Black Sholes, and just personally attacked him. In his discussion of Black Sholes, he showed how by selling puts you were effectively borrowing money at below 1%. You don't refute the results of his analysis, but only get into a petty analysis about one paragraph.

"It is clear that all of the praise that he receives from his followers has gone to his head."

I assume the "he" in this sentence is Buffett. How is Buffett's analysis of pricing in Black-Scholes make it clear to anyone (excluding petty Finance professors) that praise has gone to his head? Basically, because you don't agree with him, that proves clearly he is out of touch. I don't agree with your post. So, clearly the fact that you are a professor has gone to your head.

"He sound absurd when he talks about derivatives and seems to think that he is a macro forecaster (which actually cuts against everything he stood for two decades ago)."

First, the obvious grammatical error. Second, you think "he sound absurd" when he discusses derivatives. What could Warren Buffett possibly know about derivatives? Basically, he is absurd because you don't agree with him (Again, you don't refute his example, or show how it is wrong.) You believe his discussion is "absurd". I think it is absurd for a finance professor to post a half ass analysis, and claim it is absurd for an extremely successful derivatives investor to discuss derivatives.